# Interpreting difference in difference event study regressions

Lets say I have a model:

$$y_{i,t}= \sum_{k \neq -1} \beta_k \times treat_i \times \mathbf{1}_{K = k} + \lambda_t + \mu_i + e_{i,t},$$

where $$k$$ indicates event time, and treatment takes place at even time = $$0$$. The variable $$treat_i$$ is a dummy for treatment status, and $$\mathbf{1}_{K = k}$$ is an indicator if event time = $$k$$. These models are usually used with differences in differences to show pre-trends and trace out dynamic effects. I am wondering how you would interpret a given coefficient $$\beta_k$$ for say, $$k = 2$$:

$$(E[y|k=2,treatment]-E[y|k=2,control])- (E[y|k=-1,treatment]-E[y|k=-1,control])?$$

aka a difference in difference for the event time $$k = 2$$ compared to event time $$-1$$?

• Your interpretation looks about right. If I am correct, this model is fully saturated. It assesses all period effects before and after $k = -1$ (i.e., the omitted baseline period). Correct? Aug 7, 2020 at 0:35
• Yes that is correct. So that makes sense, so each coefficient can be interpreted like it is a difference in difference estimate for that event time relative to k= -1? Aug 7, 2020 at 0:46
• What is the difference between mu and lambda? Should one of them have an i subscript? Aug 7, 2020 at 5:43
• Yes, I corrected that in the original question. Thanks for pointing that out! Aug 7, 2020 at 5:51

Your interpretation is sufficient. Your model is fully saturated with time dummies. The omitted time dummy (i.e., $$k = -1$$) is your reference period. You could use a more distant pre-event period but most papers I have encountered use the period before treatment as the baseline.